This began with the Nintendo 64 title Pokémon Stadium, which has a special adapter for importing Game Boy save data.
Related is the specific bonuses achieved from linking a handheld with a specific game to a console with another. Mission-Pack Sequel games are especially prone to using this, as the sequel is usually built on the same basic code as the previous entry, so it's easy to port over content. Sometimes, this is a way of avoiding the Bag of Spilling problem.
Mostly, this means a tiny change (such as an acknowledgment of your accomplishments in the first game), or an item that may or may not be useful, but in some RPGs (the main originators of this) it is possible to import the character from the last game, levels intact, to start the new one (the monsters would sometimes be scaled up, of course). The ability to use old save data (or in rare cases, use of the physical game) from an old game in a new game (often a direct sequel, though unrelated examples aren't unheard of) for a bonus.
